Nonlinear Wave Equation With Infinite Memory: Analyzing Stability and Blow‐Up Under Fractional Boundary Delay

ABSTRACT This study investigates a nonlinear wave equation defined on a bounded and smooth domain , incorporating infinite memory and a nonlinear source term, together with a boundary condition that involves both a tempered Caputo fractional time‐varying delay and a linear damping effect. Local existence of solutions is established using semigroup theory, while the global existence is proved through the energy method. The exponential stability analysis is carried out by constructing a suitable Lyapunov functional and a finite‐time blow‐up result is derived when the initial energy is negative. To the best of our knowledge, this is the first study to address such a configuration within the framework of wave equations.